Some Properties of an Archimedean $\Ell $-Group

$Some Properties of an Archimedean $\Ell $-Group$

Czechoslovak Mathematical Journal

Dao Rong Tong Some properties of an archimedean ℓ-group

Czechoslovak Mathematical Journal, Vol. 45 (1995), No. 2, 293–302

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SOME PROPERTIES OF AN ARCHIMEDEAN £-GROUP

DAO-RONG TON, Nanjing

(Received March 30, 1993)

1. AUXILIARY RESULTS

We will use the standard notation for ^-groups, cf. [5]. Throughout the paper G is an ^-group, R is the real group, Q is the rational group and Z is the integer group.

If G and H are ^-groups, G ffl H denotes their cardinal sum. Let {Ga \ a G A} be a we system of ^-groups and let Yl Ga be their product. For an element g G Yl Ga aeA aeA denote the a component of g by ga. An £-group G is said to be a subdirect sum of

^-groups GQ, in symbols G C' y[ GQ, if G is an ^-subgroup of Yl Ga such that for aeA aeA each a G A and each g' G Ga there exists g G G with the property gQ = g'. An £- group G is said to be an ideal subdirect sum of ^-groups Ga, in symbols G C* Yl Ga, aeA if G C' Yl Ga and G is an £-ideal of Yl Ga. We denote the ^-subgroup of Yl Ga aeA aeA aeA consisting of the elements with only finitely many non-zero components by ^ Ga. aeA An f-group G is said to be a completely subdirect sum, if G is an ^-subgroup of

J! Ga and £ GQ C G. aeA aeA

A subset {0} ^ D C G is said to be disjoint, if Oi A g2 = 0 for any pair of distinct elements gi, g-2 € D. For any A" C G we designate X1- = {g G G | \g\ A |x| = 0 for each T G A"}. For g G G, [O] is the convex ^-subgroup of G generated by O, (g) = g11 is the polar subgroup of G generated by g. We denote the least cardinal a such that |A| ^ a for each bounded disjoint subset A of G by UG, where |A| denotes the cardinal of A. G is said to be ^-homogeneous if vH = vG for any convex ^-subgroup II ^ {0} of G. If G is an archimedean ^-homogeneous £-group and vG = K;, we call G an archimedean D-homogeneous t^-group of K; type. In [9] we proved that an ^-group G is complete if and only if G is ^-isomorphic to an ideal subdirect sum of real groups, integer groups and continuous U-homogeneous complete ^-groups. By using this result, we described the structure of an archimedean 293 £-group in [10]. Suppose that G is a subdirect sum of subgroups of reals and v- homogeneous ^-groups, G C' Yl Ts. Let Ai = {l5 G A | T^ is a subgroup of reals}. SEA If ~l Ts C G, then G is said to be a semicomplete subdirect sum of subgroups of

6eAl reals and ^-homogeneous ^-groups of Hi type, in symbols

(1.1) £ T6CGC' l\Ts. 8eAxCA 8

Theorem 1.1 (Theorem 4.7 of [10]). An E-group G is archimedean if and only if G is ^-isomorphic to a semicomplete subdirect sum of subgroups of reals and archimedean v-homogeneous i-groups of Hi type.

Now let G be an archimedean ^-group. Then we have an ^-isomorphism Q such that £ TSl C QG C' TJ TS, (5i€AiCA SeA where T$x is a subgroup of reals for each 6\ G Ai C A and Ts is an archimedean v- homogeneous ^-group of Hi type for each 6 G A \ Ai. For x G G put xl — (... x\ .. .) such that ! _ j(Qx)s ^G Ai, X& ~ jo (56 A\Ai.

We call x1 the real part of x. If for any x G G, the real part T1 G OG, G is said to be real decomposable archimedean £-group. In this case, if we put x2 = (... x2 ...) as follows: '0 «GAi, Xs 1 (QX)S SeA\Al then 1 2 QX — x -f x ,

1 and Æ2 — QX - x Є QG. Put

Gi = {Qxe QG \X EG,(QX)S =0for 6 e A\ Ai},

G2 = {DxG DG | x G G, (DT)* =0for(5G Ai}.

Then both Gi and G2 are ^-subgroups of DG, moreover,

DG = Gi fflG2.

± It is clear that G2 = R(QG) (the radical of G) and Gi = R(QG) .

294 Corollary 1.2. Let G be a real decomposable archimedean (-group. Then G is

(^-isomorphic to a cardinal sum G\ EB G2, where G\ is a completely subdirect sum of subgroups of reals and G2 is a subdirect sum of archimedean v-homogeneous £-groups of^i type.

So, if G is a real decomposable archimedean £-group, then G = R(G) ffl H(G)±. However, in general, R(G) is not a cardinal summand of G. If G is complete or laterally complete, then R(G) is a cardinal summand.

2. A COMPLETELY SUBDIRECT SUM OF SUBGROUPS OF REALS

Now we can characterize those ^-groups which can be represented as completely subdirect sums of subgroups of reals.

Theorem 2.1. Let G 7-= {0} be an archimedean (.-group. Then the following properties are equivalent: (1) G is (-isomorphic to a completely subdirect sum of subgroups of reals; (2) G is (-isomorphic to an ideal subdirect sum of real groups and integer groups; (3) G has a basis.

Proof. (1) => (2): Without loss of generality, assume

6eA seA where each Ts is a subgroup of R for S G A. Then

A G C* ft Ts\ SeA where Tf = R or Z for S € A. (2) -=> (1): It is similar to the proof of Theorem l.L (1) =-> (3): If we have the formula (1.1), then for each S (E A we choose a fixed ts with 0 < ts G Ts; the system {ts \ S € A} is a basis for G. (3) => (2): See Theorem 3.5 in [5]. •

By Theorem 4 and Corollary IV of Chapter 3 in [5] we see that an archimedean £-group G has a finite basis if and only if G is ^-isomorphic to a completely subdirect sum of a finite number of subgroups of reals. However, a completely subdirect sum of a finite number of subgroups of reals is a cardinal sum of a finite number of subgroups of reals. So we get

295 Corollary 2.2. An archimedean (-group G has a finite basis if and only if G is (.-isomorphic to a cardinal sum of a finite number of subgroups of reals.

3. HYPER-ARCHIMEDEAN PROPERTY

An £-group G is called hyper-archimedean if each £-homomorphic image of G is archimedean.

Proposition 3.1. An (-group G is hyper-archimedean if and only if G is pro tectable and [g] = (g) for each 0 < g G G.

Proof. Necessity. Suppose that G is hyper-archimedean. For any 0 < g G G we have gL ffl (g) C G. But gL ffl (g) D gx ffl [g] = G by Theorem 2.4 in [5]. So G = g1- ffl (g), and G is projectable. From G = gx ffl [g] = gx ffl (g) we get [g] = (g). Sufficiency. If G is projectable and 0 < g G G, then G = gx ffl (g). Since [g] = (g), G = gx ffl [g]. Hence G is hyper-archimedean. •

An £-group G is an a-extension of an ^-group H if and only if H is an ^-subgroup of G and the map L —)• LDH is a one-to-one map of the set of all convex ^-subgroups of G onto those of H. G is a-closed if it admits no proper a-extension.

Corollary 3.2. Let G be a hyper-archimedean (-group with a basis. If G is a-closed, then G/P ~ R for each proper prime P.

Proof. Let G be an a-closed hyper-archimedean l^-group with a basis. By the above Theorem 2.1, without loss of generality, we have

]TT, CGC' J]T,, seA seA where each Ts is a subgroup of reals. Let P be a proper prime. By Theorem 2.4 in

[5] P is maximal and P = {x G G | xs{) = 0 for some S0 G A}. So

G = TSll ffl P

and G/P ~ Ts0. If G/P0 fails to be isomorphic to R for some proper prime P0, then

G' = R ffl Fo 2 Q 3 -°o or G' = H ffl P0 2 Z ffl P0 is clearly an a-extension of G, a contradiction. Therefore G/P = H for each proper prime P. D

29G This corollary partly answers the question of the Corollary 2 in [2]. Next we discuss the hyper-archimedean kernel Ar(G) of an archimedean ^-group G. Ar(G) is a convex ^-subgroup of G which is hyper-archimedean and contains every hyper-archimedean convex ^-subgroup of G. An £-group G is continuous if for each 0 < x G G there exist x\, x2 G G such that x = x\ + x2, x\ A x2 = 0, x\ ^ 0 and #2 ^ 0.

Lemma 3.3. Let G be a complete (laterally complete and archimedean) divisible v-homogeneous £-group of Hi type. Then Ar(G) = 0.

Proof. First we can show that a projectable U-homogeneous ^-group of ttz- type is continuous. In fact, v[x] = vG = tt; for any 0 < x G G. So there exist 0 < a\ < x and 0 < a2 < x such that a\ A a2 = 0. Then G = a± ffl a.^ and so x = xi + x2 with 1 xi G a] and x2 G ar^. It is clear that x £ a± and x £ a^~. Hence x\ ^ 0, x2 ^ 0. Now let G be a complete (laterally complete and archimedean) divisible v- homogeneous ^-group of Kt- type. Then G is projectable (see [5], [4]) and continuous. Consider the Bernau representation ([3])

g: G -*G CD(XG).

For any 0 < x G G there exists a maximal disjoint subset X in G such that x G X. By Theorem 3.3 in [6] we can choose g such that x is the characteristic function of a clopen subset 5 in XQ. Since G is continuous, G is also continuous. So x = x\ + x\ with x} A rrJ = 0 and x\ ^ 0, x^ ^- 0. For 0 < xf G G we also have x\ = x\ + x'2 with x4 A x\ — 0 and X2 ^ 0, X2 7^ 0. We continue to get a sequence {xn | n = 1, 2,...} in G such that x n = XS(xi), xn A xni = 0 (71 ^ m) and l S(^) C S(x), S(x n) n 5(ai) = 0 (n -4 m), where 5(xn) is the support of xn and Xs(xl) is the characteristic function on S(xn). Put i "" G xn = — xnl and x = \\J/ ( )x„

Then xn, x G G. Now (x A nx)(t) = -^ for l G 5(xn+1). On the other hand [x A (?t + l)x](f) = 0. Therefore

x A nx = x A (n + l)x.

This proves that Ar(G) = 0 by Lemma 2.1 in [8]. •

297 Proposition 3.4. Let G be a complete (laterally complete and archimedean) v-homogeneous t-group of Hi type. Then Ar(G) = 0.

Proof. By Lemma 3.3, Ar(Gd) = 0 where Gd is the divisible hull of G. For any 0 < x G G and any n G IV we have

*]G=[^]G and 4=( П.) G'<'

G where [x]G is the convex /^-subgroup of G generated by x and [^] is the convex (.- d d subgroup of G generated by ^, XQ and (^)Gfi are polars in G and in G , respectively. Hence

By Corollary 2.1.1 in [8] we get

Ar(G) = |-| ([z]°Bxh)= H ([^"^©a^^^^0- 0<*ЄG 0

Theorem 3.5. Let G be a complete i-group. Then Ar(G) is an ideal subdirect sum of real groups and integer groups.

Proof. By Proposition 2.2 in [9], without loss of generality, we have

seA seA where each Ts (6 G A) is R or Z or a complete U-homogeneous /^-group of N; type.

Put Ai = {6 G A | Ts = R or Z}, A2 = A \ Ai. Assume x G Ar(G). For any

60 G A2 and any a^ G Ts0 we have a<,-0 = (... 0 ... as0 •. • 0 ...) G G. So there exists n £ N such that

x A nas0 = x A (n -f l)a«50.

Hence

rr^ A naSo = xSo A (n + l)a<,-0.

By Lemma 2.1 in [8] this means that x6o G Ar(T^0). However, by Proposition 3.4,

Ar(T.5()) = 0. So xSo = 0. Therefore

Ar(G) C' J] T,.

298 By Lemma 2.1. in [8] it is clear that J2 Ts Q Ar(G). Since Ar(G) is convex in G SeAr and G is convex in Yl Ts, Ar(G) is convex in Yl Ts. So we have SeAl c5GAi

£ T, C Ar(G) C* Y[ T6.

seAx seAi D

Corollary 3.6. If a complete i-group G is hyper-archimedean, then G is an ideal subdircct sum of real groups and integer groups.

Theorem 3.7. Let G be a complete (-group. Then Ar(G) is dense in G if and only if G is an ideal subdirect sum of real groups and integer groups. P r o o f. Necessity. By the proof of Theorem 3.5 we have

J2 Ts C Ar(G) C G C* J] T5 seAY seA and r Ar G r (3A) E ^ ( ) ^ n *> seAl seAi where Ts = R or Z {6

£ T, C Ar(G) C Ar(G)^ = G C* J] T,. 5€Ai 5GAi Sufficiency. Let ^ T, C G C* [] T,, (SGAj SeAi where each Ts = R or Z (S G Ai). Since

]T 7* C Ar(G) C G,

Ar(G) is dense in G. D

Corollary 3.8. Let G be a complete (-group. Then Ar(G) is dense in G if and only if G has a basis. Theorem 3.7 and Corollary 3.8 partly answer the question and conjecture in [8].

299 4. PROJECTABILITY

It is well known that a complete (cr-complete) £-group is projectable. M. An derson defined some weak concepts of projectability in [1]. An £-group G is called subprojectable if for each 0 < x G G and each non-zero polar P C x11 there exists a non-zero polar Q such that Q C P and x = Q EB Q1-. G is called densely projectable if it has a family T of non-trivial cardinal summands such that if {0} ^ P G F(G) then there exists a Q G T such that Q C P, where P(G) is the Boolean algebra of all polars in G. Suppose that H is an ^-subgroup of an £-group G. H is called a signature for GifP->PnHisa Boolean isomorphism from P(G) onto P(H). An £-group G is a specker group if it is generated as a group by its singular elements. Assume 0 < x G G. If x = x\ + #2, #i A :T2 = 0 in G, we call xi (and £2) a component of x. We call 0 ^ x G G a specker sign if for each 0 < y ^ a; there exists a non-zero component X\ of x in H11. We will say that G has a specker signature if it has a signature if it has a signature which happens to be a specker ^-subgroup. Let G be an archimedean ^-group. We denote by Ge the essential closure of G (see [6]). An element 0 < x G G is said to be saturated if, whenever there exist x\, X2 G Ge with x\ A #2 =0 in Ge such that x = x\ 4- £2, tnen #1 € G. An archimedean ^-group G is said to be saturated if each 0 < x G G is saturated. For example, a divisible complete £-group is saturated.

Proposition 4.1. A subprojectable v-homogeneous £-group G of#i type is con tinuous.

Proof. By Theorem 6 in [7] each 0 ^ x G G is a specker sign. v[x] — vG = tt2- implies that x is not basic. It follows from Lemma 9 of [7] that G is continuous. •

Proposition 4.2. A saturated archimedean (.-group is subprojectable. Proof. Let G be a saturated archimedean £-group. Consider the Bernau rep resentation

TT: G-+GC D(XG), x -> x G G.

Let 0 < y ^ x G G. By Theorem 3.3 in [6] the ^-isomorphism n can be chosen so that y is the characteristic function of a clopen subset 5 of the Stone space XQ- Put ( r S = S(x) \ S where S(x) is the support of x. Then S is also a clopen subset of XG and S(x) =5USr.

300 So we have

D(S(x)) =D(S)mD(S'),

X = X i + x2,

where xx G D(S), x2 G D(S') and D(S)(D(S')) = {/: S(S') -> (H, ±00) | / is continuous and / is real on a dense open subset of S(S')}. Since G is saturated, so is G. Hence x 1 G G. It is clear that

y% = {g G G | S($) C 5}

(see [3]). So we have xi G H^-. This proves that x is a specker sing. Hence each 0 ^ i G G is a specker sign. By Theorem 6 in [7], G is subprojectable. D

Corollary 4.3. A saturated archimedean v-homogeneous £-group of tt; type is continuous.

From Theorem 7 in [7] we have

Corollary 4.4. A saturated archimedean £-group has a specker signature.

In [1] M. Anderson proved that G is subprojectable if and only if each [x] is densely projectable. so from Proposition 4.2 we have

Corollary 4.5. Let G be a saturated archimedean (.-group. Then each [x] (x € G) is densely projectable.

Proposition 4.6. Let G be an archimedean (-group with a basis. Then G is subprojectable.

Proof. By Theorem 1.1 we have

^TaCGC' TjT,, where each Ts (S G A) is a subgroup of reals. Then for each 5 G A we choose a fixed ts with 0 < ts G Ts; the system {ts G Ts \ S G A} is a maximal disjoint subset and each ts is a specker sign (each basic is a specker sign). By 4(b) in [7], G has a specker signature. It follows from Theorem 7 in [7] that G is subprojectable. D

301 References

[1] M. Anderson: Subprojectable and locally flat latticе-orderеd groups. Univеrsity of Kansas Dissertation, 1977. [2] M. Anderson and P. Conrad: Epicomplete £-group. Algebra Univеrsalis 12 (1981), 224-241. [3] S. J. Bernau: Unique representations of archimеdеan lattice grоups and normal archimеdеan latticе rings. Prоc. Lоndоn Math. Sоc. 15 (1965), 599-631. [4] S. J. Bernau: Latеral and Dеdеkind cоmplеtiоn оf archimеdеan latticе groups. J. London Math. Soc 12 (1976), 320-322. [5] P. Conrad: Latticе-Ordеrеd Groups. Lеcture Notes, Tulane Univеrsity, 1970. [6] P. Conrad: Thе еssеntial closurе of an Archimedean lattice-ordeгed group. Dukе Matli. J. 88(1971), 151-160. [7] P. Conrad and J. Martinez: Signaturе and discrеtе latticе-ordеrеd groups. To appеar. [8] J. Martinez: Thе hypеr-archimеdean kernel sequence of a lattice-orderеd group. Bull. Austral. Math. Soc. 10 (1974), 337-349. [9] Dao-Rong Ton: Thе structure of a completе £-grоup. Czеch. Math. J. (1993). Tо appеar. [10] Dao-Rong Ton: Radical classеs оf £-grоups. Intеrnatiоiial Journal of Matliеmatics and Mathematical Science 2(17) (1994), 361-374. [11] E. C. Weinberg: Frее latticе-ordеrеd abеlian groups II. Math. Annalеn 159 (19G5), 217-222.

Author^s address: 29-305,3 Xikang Road, Nanjing, 210024, Pеoplе's Rеpublic of Cliina.

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